Place crushed candy cane bits into pint glass. Muddle the candy cane slightly, breaking up the bigger pieces. Do not over-muddle into a powder. Add vodka, brandy and half-and-half and stir. Pour into a shaker with ice. Shake lightly to chill and strain into old-fashioned glass full of ice. Add a peppermint stick and stir. Garnish with a mint sprig.
Thoroughly wash mint leaves and place in a decanter or jar. Empty vodka into container and muddle slightly to release the essential oils. Store in a cool, dark place for at least 3 days - longer if you want a stronger taste.